CD 12-1
Supplement to Chapter 12
Decision Criteria
Section 12.2 briefly introduced three decision criteria — the maximax criterion, the maximin criterion, and the maximum likelihood criterion — before focusing on Bayes’ decision rule as the criterion to be used in the remainder of the chapter. We now will describe these first three decision criteria, as well as three additional criteria, in greater detail. This presentation is self-contained and so will include much of what was said in Section 12.2 in this expanded coverage.
Recall that Max Flyer, the founder and sole owner of the Goferbroke Co., does not put much faith in the consulting geologist’s numbers in estimating that there is one chance in four of oil on the tract of land. Since these numbers led to the prior probabilities of the possible states of nature, Max would prefer to make his decision without relying on these prior probabilities if there is a good way of doing so. His daughter Jennifer, who has studied management science, agreed to begin by introducing him to some decision criteria which don’t use these probabilities before going on to others which do use probabilities.
Now let us eavesdrop on her description.
The Maximax Criterion
The maximax criterion is the decision criterion for the eternal optimist. It says to focus only on the best that can happen to us. Here is how it works.
1. For each decision alternative, determine its maximum payoff from any state of nature.
2. Determine the maximum of these maximum payoffs.
3. Choose the alternative that can yield this maximum of the maximum payoffs.
Figure 1 shows the application of this criterion to Goferbroke’s problem when using the corresponding Excel template in one of this chapter’s Excel files. It begins with the payoff table (Table 12.3) without the prior probabilities. In step 1, we find the maximum in each row (700 and 90). Step 2 identifies 700 as the maximum of these two numbers. Since 700 is the payoff that can be yielded by drilling for oil, this is the decision alternative to be chosen in step 3.
Figure 1 The Excel template for the maximax criterion, applied to the first Goferbroke Co. problem.
An Objection to This Criterion: Now suppose that the payoff table were the one shown in Figure 2. The maximax criterion again leads to choosing the alternative of drilling for oil. What a terrible decision! In the somewhat unlikely event that oil is there, drilling only does negligibly better than selling. In the far more likely event that the land is dry, drilling gives a disastrously large loss.
Figure 2 Application of the maximax criterion to a variation of the first Goferbroke Co. problem.
Max’s Reaction:
Max: At first, I thought this criterion might have possibilities because I view myself as something of an optimist. However, now I see that this criterion is really over the top in being totally optimistic. I need something that is more discriminating.
Jennifer: Now that you see that you need to rein in your optimistic tendencies, let us look at a more conservative criterion.
The Maximin Criterion
The maximin criterion is the criterion for the total pessimist. In contrast to the maximax criterion, it says to focus only on the worst that can happen to us. Here are the steps.
1. For each decision alternative, determine its minimum payoff from any state of nature.
2. Determine the maximum of these minimum payoffs.
3. Choose the alternative that can yield this maximum of the minimum payoffs.
Applying this criterion to Goferbroke’s problem gives Figure 3 (another Excel template). The basic difference from Figure 1 is that the numbers in columnH (-100 and 90) now are the minimum rather than the maximum in each row. Since 90 is the maximum of these two numbers, the alternative to be chosen is to sell the land.
Figure 3 The Excel template for the maximin criterion, applied to the first Goferbroke Co. problem.
An Objection to This Criterion: The fact that this criterion is overly cautious is dramatically illustrated in Figure 4. The maximin criterion still says to sell the land. However, the “Dry” state of nature now means there is a little oil there so drilling gives virtually the same payoff as selling. Furthermore, the “Oil” state of nature means that there is a huge oil field there. With roughly one chance in four that the latter state of nature is the true one, the gamble of drilling for oil instead is the obvious best decision.
Figure 4 Application of the maximin criterion to a variation of the first Goferbroke Co. problem.
Max’s Reaction:
Max: I can see now that this criterion would never say to drill for oil unless we absolutely knew that there was some there. That is no way to run an oil prospecting company. Don’t you have a criterion that strikes a happy medium between being totally optimistic and totally pessimistic?
Jennifer: Yes, that is our next one. You now have seen that you are neither a total optimist nor a total pessimist. This next criterion asks you to rate yourself as to just where you fall in between. On a scale from 0 to 1, where
0 = totally pessimistic,
1 = totally optimistic,
0.5 = neutral (midway between),
where would you place yourself?
Max: I definitely am a little on the optimistic side. On this scale, I would put myself at about 0.6.
Jennifer: OK, good. We call this number your pessimism-optimism index. So setting yours at 0.6, let’s now look at how this criterion works.
The Realism Criterion
The realism criterion is basically a combination of the preceding two, where the pessimism-optimism index is used to combine them appropriately. The steps are given below.
1. For each decision alternative, determine both its maximum payoff and minimum payoff from any state of nature.
2. For each decision alternative, use the pessimism-optimism index to calculate its weighted payoff as
Weighted payoff = index times maximum payoff + (1 - index) times minimum payoff.
3. Choose the alternative with the largest weighted payoff.
Figure 5 shows the application of this criterion to Goferbroke’s problem on the corresponding Excel template. Note that columns H and I are just column H of Figures 1 and 3, respectively. Column J then uses the formula in step 2, with an index of 0.6, to calculate these weighted payoffs. Since the weighted payoff for drilling (380) is larger than for selling (90), the decision is to drill for oil.
Figure 5 The Excel template for the realism criterion, applied to the first Goferbroke Co. problem.
This criterion provides a welcome middle ground between the maximax and minimax criteria. Furthermore, selecting a value for the pessimism-optimism index enables the decision maker to choose just how aggressive or cautious to be. The criterion even gives the decision maker the flexibility to be totally optimistic (index = 1) or totally pessimistic (index = 0) if desired, so the maximax and maximin criteria actually are special cases of this one.
However, this criterion also has its flaws, as indicated below.
An Objection to This Criterion: Figure 6 gives the payoff table for another problem (unrelated to Goferbroke’s problem) that has been specially designed to show an extreme case where the realism criterion performs badly. Note that alternative 2 is much better than alternative 1 for just the first state of nature, whereas the reverse is true for the other four states of nature. Assuming that the first state of nature is not particularly more likely than any of the others, alternative 1 clearly is far better than alternative 2. Nevertheless, the realism criterion with an index of 0.6 chooses alternative 2.
Figure 6 Application of the realism criterion to another example.
In fact, this criterion would choose alternative 2 with any value of the pessimism-optimism index. The reason is that this alternative has the larger value in both columns H and I of Figure 6.
Max’s Reaction:
Max: I rather like this criterion. It is more realistic than the first two criteria, and it even takes into account how optimistic or pessimistic I want to be. Furthermore, my problem only has two states of nature rather than the five in the example you just gave. Therefore, your objection to the criterion doesn’t really apply to my problem, does it?
Jennifer: Unfortunately, it does to some extent. The reason for having five states of nature in the example was to emphasize that the payoffs for one unlikely state of nature should not dictate the decision as strongly as this criterion allows.
Max: I still don’t see how this applies to my problem.
Jennifer: Well, suppose you have a tract of land that is probably dry, but there is a small possibility of a lot of oil there. This possibility is your unlikely state of nature. However, suppose the possibility is so small that it clearly is not worthwhile to drill for oil. What do you think this criterion would tell you to do?
Max: Oh oh. I suppose it would tell me to drill anywhere.
Jennifer: Yes, it would! It just doesn’t differentiate between very unlikely and somewhat likely states of nature. Now you have your consulting geologist’s report estimating that there is one chance in four of oil on your tract of land. Is this likely enough to make drilling worthwhile? This criterion just doesn’t address this question.
Max: You’re right. And this really is the key question, isn’t it? I am beginning to think that I need to use the consulting geologist’s numbers somehow, unless you have a better criterion that doesn’t need them.
Jennifer: I do have one more that you might like better.
The Minimax Regret Criterion
The minimax regret criterion gets away from the focus on optimism versus pessimism. Instead, its focus is on choosing a decision that minimizes the regret that can be felt afterward if the decision does not turn out well.
This is how regret is measured.
After observing what the true state of nature turns out to be, the regret from having chosen a particular decision alternative is
Regret = maximum payoff - actual payoff,
where maximum payoff is the largest payoff that could have been obtained from any decision alternative for the observed state of nature.
Table 1 shows the calculation of the regret for Goferbroke’s problem. On the left is the payoff table, with the maximum payoff for each state of nature given just below this table. On the right, the above formula is used with these maximum payoffs to calculate the regret for each combination of a decision alternative and a state of nature. The table on the right is called the regret table. Note that the regret is 0 if you drill for oil and oil is found, because this is the best alternative for this state of nature. The same holds true for selling the land if the land is dry. However, if you sell the land and it contains oil, you have given up a payoff of another 610 by not drilling. Similarly, drilling when the land is dry is 190 worse than selling.
Table 1 Calculation of the Regrets for the Goferbroke Co. Problem
Payoff Table / Regret TableAlternative / State of Nature
Oil Dry / Alternative / State of Nature
Oil Dry
Drill for oil / 700 -100 / 700 90
Drill for oil / -700 -(-100)
Sell the land / 90 90 / 0 190
Maximum payoff: / 700 90 / 700 90
Sell the land / -90 -90
610 0
After obtaining the regret table, the following steps are followed.
1. For each decision alternative, determine its maximum regret from any state of nature by referring to the regret table.
2. Determine the minimum of these maximum regrets.
3. Choose the alternative that can yield this minimum of the maximum regrets.
Figure 7 illustrates the application of these three steps to Goferbroke’s problem on the Excel template for this criterion. The numbers in cells H17 and H18 are obtained in step 1. Step2 determines that the minimum of these numbers is 190, so step 3 chooses the corresponding alternative of drilling for oil. This alternative guarantees that the regret after learning the true state cannot exceed 190, whereas the regret can be as large as 610 with the other alternative.
CD 12-12
Figure 7 The Excel template for the minimax regret criterion, applied to The first Goferbroke problem.
CD 12-12
An Objection to This Criterion: Now let us add a third decision alternative to the problem. Based on the consulting geologist’s report, an insurance company would be willing to sell the Goferbroke Co. an insurance policy to protect against the land being dry. If Goferbroke pays a massive premium and then drills for oil without finding any, the insurance company will pay an amount $700,000 larger than the premium. After deducting the cost of $100,000 for drilling, this would leave a profit of $600,000 if the land is dry. Unfortunately, the premium for this insurance policy is so exorbitant — $6.7 million—that Max could never buy the policy. Even if he finds oil for a gain of $700,000, the net loss of $6 million would put him out of business immediately.
Nevertheless, let us go ahead and apply the minimax regret criterion when the bad alternative of buying the insurance policy is included in the problem. Continuing to use units of thousands of dollars, the payoff table for this problem is shown in the top half of Figure 8.
Figure 8 Application of the minimax regret criterion to the first Goferbroke Co. problem when an insurance option is included.
The maximum payoff for each state of nature given in row 11 is used to calculate the regrets. This yields the regret table shown in rows 14-21. Applying the minimax regret criterion to this regret table presumably will lead either to choosing the new alternative, buy insurance, or the alternative that was chosen before, drill for oil (see Figure 7). Right? Wrong! For some reason, introducing a new alternative that is soundly rejected leads this criterion to switch its choice to the alternative that was rejected in Figure 7.